MATH 669: COMBINATORICS OF POLYTOPES Alexander Barvinok Abstract. These are rather condensed notes, not really proofread or edited, pre- senting key definitions and results of the course that I taught in Winter 2010 term. Problems marked by ◦ are easy and basic, problems marked by ∗ may be difficult. Typeset by AMS-TEX 1 Contents 1. Basic definitions 3 2. Carath´eodory’s Theorem 4 3. Radon’s Theorem 5 4. Helly’s Theorem 6 5. Euler characteristic 7 6. Polyhedra and linear transformations 10 7. Minkowski sum 12 8. Examples of valuations 14 9. The structure of polyhedra 16 10. The Euler-Poincar´eformula 23 11. The Birkhoff polytope 24 12. The Schur-Horn Theorem 26 13. Transportation polyhedra 27 14. The permutation polytope 28 15. Cyclic polytopes 31 16. Polarity 32 17. Polytopes and polarity 36 18. Regular triangulations and subdivisions 38 19. The secondary polytope 39 20. Fiber polytopes 43 21. Identities modulo polyhedra with lines 46 22. The exponential valuation 49 23. A formula for the volume of a polytope 54 24. Simple polytopes and their h-vectors 55 25. The Upper Bound Theorem 58 26. Balinski’s Theorem 61 27. Reconstructing a simple polytope from its graph 63 28.Thediameterofthegraphofapolyhedron 65 29. Edges of a centrally symmetric polytope 66 30. Approximating a convex body by an ellipsoid 68 31. Spherical caps 70 32. An inequality for the number of faces of a centrally symmetricpolytope 72 33. Gale transforms and symmetric Gale transforms 75 34. Almost Euclidean subspaces of ℓ1 and centrally symmetric polytopes with many faces 77 35. The volume ratio and almost Euclidean subspaces 79 2 1. Basic definitions We work in a finite-dimensional real vector space V . Once we choose a basis of V , we may identify V = Rd. (1.1) Definitions. For two points a,b V , we define the interval [a,b] V by ∈ ⊂ [a,b] = x V : x = λa + (1 λ)b : 0 λ 1 . ∈ − ≤ ≤ n o A set X V is convex if for all a,b X we have [a,b] X. The empty set is convex. Given⊂ points, a ,...,a V ∈, a point ⊂ ∅ 1 n ∈ n n a = λ a where λ = 1 and λ 0 for i =1,...,n i i i i ≥ Xi=1 Xi=1 is called a convex combination of a1,...,an. The convex hull conv(A) of a set A V is the set of all convex combinations of points from A. A polytope is the convex⊂ hull of a finite set of points. Given a linear functional ℓ : V R, not identically 0, and a real α R, the set −→ ∈ H = x V : ℓ(x) α − ∈ ≤ n o is called a (closed) halfspace. A polyhedron is the intersection of finitely many halfspaces. (1.2) Problems. 1◦. Prove that conv(A) is the minimal under inclusion convex set containing A. 2◦. Prove that conv(conv(A)) = conv(A), that conv(A) conv(B) provided A B and that conv(A) conv(B) conv(A B). Prove⊂ that if u / conv(A), v /⊂conv(A), u conv A ∪ v and v⊂ conv A∪ u then u = v. ∈ ∈ ∈ ∪{ } ∈ ∪{ } 2 3. Let us identify C = R . Let f : C C be a polynomial. Prove that the zeros of the derivative f ′ of f lie in the convex−→ hull of the zeros of f (Gauss - Lucas Theorem). ◦ d 4 . Let e1,...,ed be the standard basis of R . Let us define: ∆ = conv (e ,...,e ) , O = conv ( e ,..., e ) and d−1 1 d d ± 1 ± d I = conv ( e e ... e ) . d ± 1 ± 2 ± d Prove that ∆d−1, Od, and Id are polyhedra and find the minimal set of linear inequalities defining each. Polytope ∆ is called the standard (d 1)-dimensional simplex, polytope O d−1 − d is called the standard d-dimensional octahedron or cross-polytope and polytope Id is called the standard d-dimensional cube. 3 5∗. Prove that a polytope is a polyhedron and a bounded polyhedron is a polytope (Weyl - Minkowski Theorem, to be proven later). 6∗. Prove that the set of diagonals of n n symmetric matrices with the eigen- × n values λ1,...,λn is the convex hull of the vectors in R whose coordinates are permutations of λ1,...,λn (Schur-Horn Theorem, we will prove at least a part of it later). 2. Caratheodory’s´ Theorem (2.1) Theorem. Let dim V = d and let A V be a set. Then every point x conv(A) is a convex combination of some d +1⊂ points from A. ∈ Proof. Let us choose a point x conv(A). Then x is a convex combination of some points from A, ∈ n n x = λiai where λi =1 i=1 i=1 X X and where without loss of generality we may assume that λi > 0 for i =1,... ,n. If n d + 1 we are done, since we can append the combination by 0s as needed. It suffices≤ to prove that if n>d + 1 then we can represent x as a convex combination of fewer ais. Let us consider a homogeneous system of linear equations in real variables α1,...,αn: n n αiai = 0 and αi =0. Xi=1 Xi=1 The number of equations is d + 1, so there is a non-trivial solution α1,...,αn. In particular, for some i we must have α > 0. For t 0 let us define i ≥ λ (t) = λ tα for i =1,... ,n. i i − i Clearly, n n x = λi(t)ai and λi(t)=1. i=1 i=1 X X Let us choose λ t = min i . i: αi>0 αi Then λi(t) 0 and for some j we have λj(t) = 0. Hence we expressed x as a convex combination≥ of fewer points. 4 (2.2) Problems. 1. Suppose that the set A is path-connected. Prove that every point x conv(A) is a convex combination of some d points from A. ∈ ∗ 2 . Let A1,...,Ad+1 V be sets, where dim V = d. Suppose that x conv (A ) for i = 1,...,d ⊂+ 1. Prove that there exist points a A such that∈ i i ∈ i x conv (a1,...,ad+1). This is the “colored Carath´eodory Theorem” proved in I. B´ar´any,∈ A generalization of Carath´eodory’s theorem, Discrete Math. 40 (1982), no. 2-3, 141–152. 3. Radon’s Theorem (3.1) Theorem. Let dim V = d and let A V be a set with at least d +2 points, A d + 2. Then one can find subsets ⊂R,B A such that R B = and conv(| | ≥R) conv(B) = . ⊂ ∩ ∅ ∩ 6 ∅ Proof. Let a ,...,a , n d + 2 be some distinct points from A. Let us consider 1 n ≥ the following system of homogeneous linear equations in real variables β1,...,βn: n n βiai = 0 and βi =0. i=1 i=1 X X The number of equations is d + 1 and since n>d + 1 there exists a non-trivial solution to the system. In particular, for some i we have βi > 0 and for some j we have βj < 0. Let γ = β = ( β ) > 0. i − i i:Xβi>0 i:Xβi<0 Let R = ai : βi > 0 and B = ai : βi < 0 . Clearly, R B = andn for the pointo n o ∩ ∅ β β p = i a = i a γ i − γ i i:Xβi>0 i:Xβi<0 we have p conv(R) conv(B). ∈ ∩ (3.2) Problem. 1∗. Suppose that dim V = d and that A V is a set such that A (d + 1)(k 1) + 1 for some integer k. Prove⊂ that one can find pairwise disjoint| | ≥ subsets A−,...,A A such that 1 k ⊂ conv (A ) ... conv (A ) = . 1 ∩ ∩ k 6 ∅ This is Tverberg’s Theorem, see H. Tverberg and S. Vre´cica, On generalizations of Radon’s theorem and the ham sandwich theorem, European J. Combin. 14 (1993), no. 3, 259–264 for a relatively intuitive proof. 5 4. Helly’s Theorem (4.1) Theorem. Suppose that dim V = d and let S1,...,Sm V be convex sets such that ⊂ S ... S = for all 1 i < i <...<i m. i1 ∩ ∩ id+1 6 ∅ ≤ 1 2 d+1 ≤ Then m S = . i 6 ∅ i=1 \ Proof. The proof is by induction on m. The case of m = d + 1 is tautological. Suppose that m>d + 1. By the induction hypothesis, the intersection of every (m 1) of sets Si is not empty. Therefore, for i =1,...,m, there is a point pi such that− p S provided j = i. i ∈ j 6 If some two points pi1 and pi2 coincide, we will have constructed a point p = pi1 = pi2 which belongs to every set Si. If the points p1,...,pm are distinct, we apply Theorem 3.1 to the set A = p1,...,pm and claim that there are disjoint subsets R,B A such that conv(R{) conv(B}) = . Let us choose a point p ⊂ ∩ 6 ∅ ∈ conv(R) conv(B). Let us consider a set Si. Then either pi / R or pi / B. If p / R, we∩ have R S and hence p S since S is convex.∈ If p / B, we∈ have i ∈ ⊂ i ∈ i i i ∈ B Si and hence p Si since Si is convex. In either case, p Si for i =1,...,m. ⊂ ∈ ∈ (4.2) Problems. ◦ 1 . Let Si : i I be a possibly infinite family of compact convex subsets S V , dim{V = d,∈ such} that the intersection of every d+1 of sets S is non-empty. i ⊂ i Prove that the intersection of all sets Si is non-empty.